Multlipication and Division of Algebraic Fractions – JSS2 Mathematics Lesson Note (First Term)

Algebraic fractions are fractions whose numerators or denominators contain letters as well as numbers. To add or subtract them, change them to equivalent fractions with the same denominator, using the LCM. To multiply, cancel common factors and multiply across. To divide, change ÷ to × and turn the second fraction upside down.

Behavioural objectives

By the end of the lesson, students should be able to:

• add and subtract algebraic fractions with different denominators using the LCM;
• reduce an algebraic fraction to its simplest form by dividing out common factors;
• multiply and divide algebraic fractions;
• simplify algebraic fractions that have brackets in the numerator.

How do you add or subtract algebraic fractions?

To add or subtract fractions with different denominators, first change them to equivalent fractions. This is done by finding the LCM of the denominators.

Example 1

Simplify 2a/5 + 4a/3

Solution:

The LCM is 15.

(6a + 20a)/15 = 26a/15

Example 2

Simplify 3/(5x) + 1/(2x) – 1/(4x)

Solution:

The LCM of the denominators is 20x.

(12 + 10 – 5)/20x = 17/(20x)

How do you simplify (reduce) an algebraic fraction?

Divide the numerator and the denominator by their common factor.

Example 1

Reduce 25x⁴y³/35x³y³

Solution:

Divide through by 5x³y³ (the common factor).

= 5x/7

Example 2

Reduce 8x³y²/6x³z

Solution:

Divide through by 2x³ (the common factor).

= 4y²/3z

How do you multiply and divide algebraic fractions?

To multiply, cancel common factors and multiply across. To divide, change the division sign to multiplication and turn the second fraction upside down.

Example 1

Simplify (x – 2)/7 × 4/(x – 2)

Solution:

(x – 2)/7 × 4/(x – 2): the two (x – 2) terms cancel each other.

= 4/7

Example 2

Simplify 6x²/11y ÷ 18x/33y²

Solution:

Change the division sign to multiplication and turn the second fraction upside down:

6x²/11y × 33y²/18x

= xy

How do you simplify fractions with brackets?

Simplify the following: (a) (2x + 5)/4 + (2x – 3)/4 (b) (7x – 2)/4 + (x – 4)/6

Solution (a)

The LCM is 4.

[(2x + 5) + (2x – 3)]/4

= (4x + 2)/4 = 2(2x + 1)/4

= (2x + 1)/2

Solution (b)

The LCM is 12.

[3(7x – 2) + 2(x – 4)]/12

= (21x – 6 + 2x – 8)/12

= (23x – 14)/12

Evaluation

1. Simplify 3a/4 + 2a/3.
2. Reduce 12x³y²/18x²y.
3. Simplify (x + 3)/5 × 10/(x + 3).
4. Simplify 4x²/9y ÷ 8x/27y².

Assignment

1. Simplify 3b/4 + 5b/6.
2. Reduce 15x³y²/20x²y³.
3. Simplify (2x + 1)/3 + (x – 4)/6.